Circle tangent question?

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Answer 1 · 2018-04-23T00:29:12

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Let K the tangent RST intersects the smaller circle at K.

Construction

$O, Q; O, S and Q, T$ $O,Q;O,SandQ,T$ are joined.

Given ST is tangent to the circle of center O and $∠ Q S T = 46^{\circ}$ $angle QST=46^@$

We have alternate $∠ Q P S = ∠ Q S T = 46^{\circ}$ $angleQPS=angle QST=46^@$

Hence central $∠ Q O S = 2 ∠ Q P S = 92^{\circ}$ $angleQOS=2angleQPS=92^@$

Now in cyclic quadrilateral $O Q T S$ $OQTS$

Sum of opposite angles

$∠ Q O S + ∠ Q K S = 180^{\circ}$ $angleQOS+angleQKS=180^@$

$\Rightarrow 92^{\circ} + ∠ Q K S = 180^{\circ}$ $=>92^@+angleQKS=180^@$

$\Rightarrow ∠ Q K S = 180^{\circ} - 92^{\circ} = 88^{\circ} < 90^{\circ}$ $=>angleQKS=180^@-92^@=88^@<90^@$

So $∠ Q K S < 90^{\circ}$ $angleQKS<90^@$ can not be semicircular angle.

Hence $S Q$ $SQ$ is not a diameter.